a 
( 634 ) 
6. If we bring the vertex S of the cone [S| on the surface of 
[FR], then 7 coincides with S, whilst the point S= 7’ becomes a 
node of r*; so 4* possesses besides the two cusps a node and in 
this point S' and 7” lie united. Out of each cusp only one tangent 
more can be drawn to 4*, and these two tangents intersect each 
other in the point R’ lying on h. Through this point pass two more 
tangents to 4*, the projections of the two generatrices of the cone 
(R] touching r*. The points of contact of these two tangents with 
7 lie in the plane e (§ 4), the common polar plane of A with 
respect to the cones {S| and [MZ], and so on the conie r? which 
has this plane in common with the cone {[#]. This conic contains 
the vertex S; so now k* is harmonically situated with respect to the 
point R' and the conic r* passing through the double point and the 
points of contact of the two tangents out of R' not passing through 
the cusps, and touching the tangents out of R' which do pass through 
the cusps in the harmonically conjugated points of R' with respect to 
the cusps and the points of contact. 
If we bring through the line HA an arbitrary plane u, (see $ 4), 
then the harmonically conjugate u, always coincides with the tangential 
plane HRS to [H|, which plane contains no other point of r* than 
the node; of each group considered in $4 of 8 points there are four 
coinciding in the node, whilst the four remaining ones form a complete 
quadrangle with the diagonal points #, H, R*. The tangents to r* in 
two points on a straight line through /# intersect each other in 9g and the 
locus of this point of intersection is a plane &* containing the node 
of r*, having in this point a cusp (with cuspidal tangent in the plane 
RST), passing through H and having with #* two points in common, 
whose tangents pass through &. So if we pass to the central 
projection of r*, we find that by the rays of the pencil (R') on kt 
again a quadratic involution is generated in such a way that the two 
points of each pair form with R’ and one of the two points of inter- 
section of the ray under discussion with r’? a harmonic group; the 
point of intersection of the tangents in the points of a pair moves 
along a cubic curve lying harmonically with respect to h and H, 
containing the cusps of k* and touching here the cuspidal tangents, 
passing through the points of contact of the two tangents out of R’ 
not passing through the cusps, passing through H and having in the 
node of k* a cusp with cuspidal tangent h. 
The cusps, the point of contact of the two tangents out of A’ 
and the node represent all the twelve points of intersection of the 
cubic curve with 4f. 
